Image reconstruction method for collimator and detector based medical imaging systems

ABSTRACT

A method includes providing a target object and acquiring measured images of the target object. Each of the measured images is acquired by filtering radiation from the target object by a mask having multiple holes and detecting filtered radiation by a detector. The method further includes providing an estimated image of the target object and calculating an updating factor for each of the measured images. The calculating of the updating factor includes partitioning a mathematical representation of the mask into multiple first regions; for each of the first regions, deriving a separate forward projection from the estimated image of the target object and the respective first region; and comparing the respective measured image of the target object with the forward projections. The method further includes updating the estimated image of the target object based on the updating factors.

PRIORITY

This application is a continuation of U.S. patent application Ser. No.16/792,672 filed on Feb. 17, 2020, which claims priority to U.S.Provisional Patent Application Ser. No. 62/807,106 filed on Feb. 18,2019, both of which are incorporated herein by reference in theirentireties.

BACKGROUND

In molecular medical imaging, sometimes known as nuclear medicine,images representing radiopharmaceutical distributions may be generatedfor medical diagnosis. Prior to imaging, radiopharmaceuticals areinjected into an imaging subject such as a patient. Theradiopharmaceuticals emit radioactive photons, which can penetratethrough the body to be detected by a photon detector. Based oninformation from the received photons, the photon detector may thendetermine the distribution of the radiopharmaceuticals inside thepatient. Their distribution represents the physiological function of thepatient, and therefore images of their distribution provide valuableclinical information for diagnosis of a variety of diseases andconditions such as those in cardiology, oncology, neurology, etc.

A collimator is a device that guides photon path. In molecular imaging,photons may originate from unknown locations inside a subject, unlike inX-ray or CT where photons are emitted from a known source (or sources)position. Without collimators, photons from all directions may berecorded by gamma detectors, and image reconstruction may becomedifficult. Therefore, collimators are employed to guide possible photonpaths so that images can be reconstructed, similar to the role of lensin a photography camera. Although existing collimator and detectorimaging systems have been generally adequate for their intendedpurposes, they have not been entirely satisfactory in all respects. Forexample, existing imaging systems are often limited by background noiseand nonuniformity artifacts. Therefore, improvement on imagereconstruction methods is desired to increase imaging sensitivity orresolution for collimator and detector based medical imaging systems.

SUMMARY

According to various embodiments, the present disclosure provide amethod of imaging reconstruction, including providing a target object, adetector, and a mask disposed between the target object and thedetector; acquiring a measured image of the target object by thedetector; providing an estimated image of the target object;partitioning the mask into multiple regions; for each of the regions,deriving a forward projection from the estimated image of the targetobject and the respective region, thereby acquiring multiple forwardprojections; comparing the measured image of the target object with theforward projections; and updating the estimated image of the targetobject based on a result of the comparing. In some embodiments, themethod further includes repeating the steps of deriving, comparing, andupdating. In some embodiments, the method further includes on conditionthat a difference of the estimated images of the target object inconsecutive two steps is less than a predetermined threshold, storingone of the estimated images of the target object as a reconstructedimage of the target object. In some embodiments, the method furtherincludes on condition that the step of updating the estimated image ofthe target object is repeated for a predetermined number of times,storing the estimated image of the target object as a reconstructedimage of the target object. In some embodiments, the method furtherincludes repeating the steps of partitioning, deriving, comparing, andupdating, wherein a number of the regions increases during the repeatingof the steps. In some embodiments, the deriving of the forwardprojection includes a convolution operation. In some embodiments, thederiving of the forward projection includes for each of the regionscalculating a respective angular effect correction factor. In someembodiments, the angular effect correction factor includes a cos³(θ)term, θ being an incident angle. In some embodiments, the mask hasmultiple through holes, and wherein each of the regions has at least onethrough hole. In some embodiments, the mask has at least two regionshaving different numbers of through holes. In some embodiments, each ofthe regions has a convex shape. In some embodiments, updating theestimated image includes for each of the regions calculating a backwardprojection based on applying a respective forward projection to acorrelation operation.

According to various embodiments, the present disclosure also provides amethod of imaging processing, including providing a target object and amask partially blocking a radiation from the target object; providing anestimated image of the target object; partitioning the mask intomultiple regions; for each of the regions, deriving a forward projectionfrom the estimated image of the target object and the respective region,thereby acquiring multiple forward projections. wherein the deriving ofthe forward projection includes for each of the regions calculating arespective angular effect correction factor. In some embodiments, theangular effect correction factor includes a cos³(θ) term, θ being anincident angle. In some embodiments, the method further includesacquiring a measured image of the target object by a detector; comparingthe measured image of the target object with the forward projections;and updating the estimated image of the target object based on a resultof the comparing.

According to various embodiments, the present disclosure also provides amedical imaging system, including a collimator configured to filterradiation emitted from a target object; a detector configured to acquirea measured image of the target object by detecting the radiation thathas passed through the collimator; and a controller operable to executecomputer-readable codes to perform following operations: receiving themeasured image from the detector; providing an estimated image of thetarget object; partitioning the collimator into multiple regions; foreach of the regions, deriving a forward projection, thereby acquiringmultiple forward projections; and updating the estimated image based ona result of comparing the measured image and the forward projections. Insome embodiments, the initial estimated image is acquired from a CTscan. In some embodiments, steps of deriving the forward projection andupdating the estimated image are part of an iteration operation. In someembodiments, the step of partitioning the collimator is also part of theiteration operation, and wherein a number of the regions increasesduring the iteration operation. In some embodiments, for each of theregions, there is one or more through holes forming a coded aperturepattern.

According to various embodiments, the present disclosure also provides acollimating apparatus for medical imaging, including a perforated platewith a top surface and a bottom surface; and a plurality of holes,wherein each of the plurality of holes extends from the top surface tothe bottom surface, wherein: the plurality of holes is grouped into twoor more regions of the perforated plate; and each of the two or moreregions includes a coded aperture pattern formed by a portion of theplurality of holes. In some embodiments, the coded aperture pattern isone of a URA array, a MURA array, a random array, and a pseudo randomarray. In some embodiments, the coded aperture pattern in each of thetwo or more regions is different from each other.

BRIEF DESCRIPTION OF THE DRAWINGS

The present disclosure is best understood from the following detaileddescription when read with the accompanying figures. It is emphasizedthat, in accordance with the standard practice in the industry, variousfeatures are not drawn to scale and are used for illustration purposesonly. In fact, the dimensions of the various features may be arbitrarilyincreased or reduced for clarity of discussion.

FIG. 1 is a schematic diagram of an exemplary nuclear imaging systemaccording to various aspects of the present disclosure.

FIGS. 2A and 2B are schematic top and cross-sectional views,respectively, of an exemplary collimator according to various aspects ofthe present disclosure.

FIGS. 3A and 3B are a schematic top view and corresponding matrix of anexemplary partitioned collimator according to various aspects of thepresent disclosure.

FIGS. 4A, 4B, and 4C are exemplary embodiments in partitioning regionson a collimator according to various aspects of the present disclosure.

FIG. 5 is a flow chart of a method of image reconstruction according tovarious aspects of the present disclosure.

FIG. 6 is an exemplary collimator design having a plurality of regionstherein, where each region includes a coded aperture pattern, accordingto various aspects of the present disclosure.

FIG. 7 is an exemplary collimator design having a plurality of regionstherein, where one region has a rotated pattern with respect to patternsin other regions.

DETAILED DESCRIPTION

The following disclosure provides many different embodiments, orexamples, for implementing different features of the disclosure.Specific examples of components and arrangements are described below tosimplify the present disclosure. These are, of course, merely examplesand are not intended to be limiting. Any alterations and furthermodifications to the described devices, systems, methods, and anyfurther application of the principles of the present disclosure arefully contemplated as would normally occur to one having ordinary skillin the art to which the disclosure relates. For example, the features,components, and/or steps described with respect to one embodiment may becombined with the features, components, and/or steps described withrespect to other embodiments of the present disclosure to form yetanother embodiment of a device, system, or method according to thepresent disclosure even though such a combination is not explicitlyshown. In addition, the present disclosure may repeat reference numeralsand/or letters in the various examples. This repetition is forsimplicity and clarity and does not in itself dictate a relationshipbetween the various embodiments and/or configurations discussed.

Moreover, a feature on, connected to, and/or coupled to another featurein the present disclosure that follows may include embodiments in whichthe features are in direct contact, and may also include embodiments inwhich additional features may interpose the features, such that thefeatures may not be in direct contact. In addition, spatially relativeterms, for example, “lower,” “upper,” “horizontal,” “vertical,” “above,”“over,” “below,” “beneath,” “up,” “down,” “top,” “bottom,” etc., as wellas derivatives thereof (e.g., “horizontally,” “downwardly,” “upwardly,”etc.) are used for ease of the present disclosure of one featuresrelationship to another feature. The spatially relative terms areintended to cover different orientations of the device including thefeatures. Still further, when a number or a range of numbers isdescribed with “about,” “approximate,” and the like, the term isintended to encompass numbers that are within a reasonable rangeincluding the number described, such as within +/−10% of the numberdescribed or other values as understood by person skilled in the art.For example, the term “about 5 mm” encompasses the dimension range from4.5 mm to 5.5 mm.

The present disclosure is generally related to the field of medicalimaging, and more particularly to an imaging reconstruction method tocollimator and detector based medical imaging systems, such as to singlephoton emission computerized tomography (SPECT) or positron emissiontomography (PET) based on near-field coded aperture collimation andmaximum likelihood estimation in some embodiments.

Prior to taking molecular medical images, a radiopharmaceutical isusually taken orally or injected into the patient. Theradiopharmaceutical undergoes nuclear decay, emitting, either directlyor indirectly through annihilation, gamma photons at certain rates andwith characteristic energies. One or more detector unit are placedaround the patient or object to record or monitor emissions. In manycases, for convenience of manufacturing and data processing, thedetectors are organized in planar shape, therefore acquire data in 2Dmatrix format, which are often referred to as projections. Based on therecorded information including position, energy and counts of suchdetected events, an image of the radiopharmaceutical distribution can bereconstructed to study the function of certain body parts.

FIG. 1 illustrates an exemplary molecular or nuclear imaging system 10,which may be used to medically examine or treat a subject such as apatient. In one embodiment, imaging system 10 is a SPECT imaging system.Alternatively, imaging system 10 can be other molecular or nuclearimaging systems, such as a PET imaging system. For the sake ofsimplicity, a SPECT imaging system will be illustrated as an example todemonstrate image reconstruction methods. However, a person in thepertinent field will understand the proposed image reconstructionmethods are not limited to a SPECT imaging system, but can be applied toother suitable imaging systems, such as a PET imaging system. Imagingsystem 10 includes an imaging apparatus 100, a gantry 110, a platform112, a control console 120, and a computer system 130. In theillustrated embodiment, computer system 130 includes a data storage unit132, an image processor 134, an image storage unit 136, and a display138. Imaging apparatus 100 is mounted on gantry 110, which may move,rotate, and acquire data. Patient 150 is placed on platform 112 (e.g., acouch) for examination or treatment by the imaging apparatus 100. Insome embodiments, imaging apparatus 100 is coupled to gantry 110 throughmovable parts so that they may move (e.g., rotate) on gantry 110.

Imaging apparatus 100 detects and records radiation emitted from patient150 and transfers recorded information to data storage unit 132. Then,image processor 134 may use the recorded information to reconstructvolumetric images representing radiopharmaceutical distributions withinpatient 150. The reconstructed images are stored in image storage unit136, which can be manipulated and displayed on display 138 for viewing.Control console 120 may be used by an operator or technician to controlimaging apparatus 100 in acquiring data. In some embodiments, controlconsole 120, data storage unit 132, image processor 134, image storageunit 136, and display 138 are integrated in a computer system 130. Insome embodiments, one or more computer components (such as controlconsole 120, data storage unit 132, image processor 134, image storageunit 136, and display 138) can be partially or entirely located at aremote location (e.g., on the cloud). In some embodiments, one of moreof these components may exist locally or remotely.

Imaging apparatus 100 includes detector 102 and collimator (orcollimators) 104. In some embodiments, detector 102 is a semiconductordetector, such as one based on cadmium telluride (CdTe), cadmium zinctelluride (CZT), or high purity germanium (HPGe). In some embodiments,detector 102 is a scintillator (such as sodium iodide (NaI) or caesiumiodide (CsI) based) detector. In some other embodiments, detector 102may also be a scintillator coupled with compact photo multiplier tubes(PMTs), silicon photomultiplier tubes (SiPMT), or avalanche photodiodes.Collimator 104 includes one or more openings, such as through holes.Depending on number and geometrical placement of through holes,collimator 104 may be a single-pinhole, multi-pinhole, coded aperture,or extended coded aperture (also known as spread field imaging, SFI)collimator, or other suitable types of collimator. Depending on profilesof through holes, collimator 104 may be a parallel-hole, fan-beam, orcone-beam collimator, or other suitable types of collimator.

One or more radiopharmaceuticals orally taken or injected into patient150 undergo nuclear decay and may emit, either directly or indirectlythrough annihilation, radiation (e.g., gamma photons) at certain ratesand with characteristic energies. Detector 102 is placed near patient150 to record or monitor emissions. Based on recorded information suchas position, energy, and counts of such detected events, an image ofradiopharmaceutical distribution may be reconstructed to study thestatus or function of certain body parts on patient 150. In SPECTimaging, collimator 104 is placed between detector 102 and an imagingobject, the openings on the collimators determining the directions andangular span from which radiation can pass through to reach certainposition on the detector.

In various embodiments, collimators are essentially perforated platesusually made of heavy metal such as lead and tungsten. In someembodiments, the collimator is made of planar plates, usually placed inparallel to the planar detector surface. The thickness of the plate,depending on the energy of photons it is designed to imaging, is largeenough to stop the majority of the radiation so that the photonsprimarily pass through the small pinholes on the plate. For example, forthe commonly used isotope, Technetium-99m (99mTc), emitting gamma rayswith energy around 140 keV, a 3 mm thickness is usually enough for aplate made of lead, and about 2 mm for tungsten. The thickness needs tobe greater to image higher energy gamma rays. These collimators need tobe placed at certain distance from the detector to allow photons comingfrom the design field-of-view (FOV) passing the pinhole(s) to spreadacross the detector surface. A gap between a collimator and a detectorin this scenario is usually greater than 3 cm.

Imaging apparatus 100 may include other necessary parts for an imaginggantry such as connectors that couple parts together (e.g., connectingdetector 102 and collimator 104 together), motors that cause parts tomove, photon shielding components, a housing component that containsother parts, etc. For example, a coupling and shielding component 106may connect detector 102 and collimator 104 such that both move (e.g.,rotate) together, and prevent radiation (photons) from reaching detector102 through paths other than collimator 104. In other embodiments,detector 102 and collimator 104 may move individually with respect toeach other.

FIGS. 2A and 2B illustrate a top view and a cross-sectional side view ofan exemplary collimator 104. Collimator 104 is configured to filterradiation by blocking certain photons and passing through other photons.Collimator 104 is made of radiation (e.g., photons) absorbing heavymetal(s), such as lead and/or tungsten. Collimator 104 has openings 108built therein to allow some photons to pass through and reach detector102 (FIG. 1 ). It should be understood that radiation or photon blockingor absorption by a collimator does not require blocking of 100% ofphotons because a small percentage of photons (e.g., 5% or less) maystill penetrate through the full thickness of the radiation absorbingmaterial. The number of escaping photons may decrease exponentially withthe thickness of a collimator. In other words, blocking (or othersimilar terms) means that substantially all of the photons (e.g., 95% ormore, or 99% or more) are absorbed by the radiation absorbing material.

Openings 108 may be through holes that extend from a top surface of thecollimator to a bottom surface of the collimator. Alternatively,openings 108 may be substantially through the collimator—such as beingrecesses from a top surface of the collimator with a depth of over 98%of a thickness of the collimator. Openings 108 may also be calledthrough holes, tunnels, apertures, or pass-through features—may have anysuitable shape, size, number, and/or distribution within theirrespective collimators. In some embodiments, openings 108 may includeparallel holes, fan beams, cone beams, slit-slat, pinholes,multi-pinholes, coded aperture, any other suitably shaped openings, orcombinations thereof. In some embodiments, collimator 108 is placedclose (e.g., 2 cm or less) to patient 150. Thus, collimator 108 may useparallel holes or fan-beams (converging or diverging) since suchfeatures do not need significant separation from patient 150. In someembodiments, openings 108 may be slanted, converging, or diverging andmay form fan beams or cone beams, etc. In an example, openings 108include a plurality of pinholes, where the number of pinholes may begreater than 11, greater than 23, or greater than 59, or greater than100. Openings 108 may form a coded aperture pattern, for example, anMURA (modified uniformly redundant array) of sizes 5, 7, 11, and 13comprise 12, 24, 60, and 84 holes, respectively. A higher number ofpinholes helps improve imaging sensitivity. Further, openings 108 may besingle pinhole, multi-pinhole, multiple pinhole modules (includingspread field imaging (SFI) or coded aperture).

As shown in FIG. 2B, a radiation source (e.g., from a part of patient150) 152 emits photons from above collimator 104. Photons hit a topsurface of collimator 104 will be blocked. Photons passing through a topside (the side facing the radiation source) of opening 108 may still beblocked by its side walls. An incident path 154 of a photon hitting anedge point A of the top opening of opening 108 forms an angle θ withrespect to vertical direction Z that is the normal of the top surface ofcollimator 104. The angel θ is referred to as an incident angle. Whenincident angle θ is not equal to 0 degree (θ≠θ), the radiation source152 is termed an off-center source. The incident angle θ is consideredsubstantially the same when the incident path 154 is measured fromanother point of opening 108, such as measured from an edge point B atthe bottom side of opening 108, or any point between edge points A andB, or a point in close proximity to opening 108 if the thickness andopening are substantially smaller than the distance between the source152 and the opening 108.

If a photon travels towards collimator 104 along an incident path withan angle larger than the incident angle θ, the photon would be absorbedby collimator 104 (note there are occasions where the photon cutsthrough a portion of collimator 104 adjacent the opening (e.g., a thinarea on the sidewall of the opening)). In some embodiments, the incidentangle θ ranges from 0° to about 2° or from 0° to about 10°. In anexample, a LEHR (low energy high resolution) collimator has an openingdiameter of about 1.11 mm and a length of about 24.04 mm, with anacceptable incident angle range of 0° to about 2.64°. In anotherexample, a GAP (general all purpose) collimator has an opening diameterof about 1.40 mm and a length of about 25.4 mm, with an acceptableincident angle range of 0° to about 3.15°. In yet another example, aLEHS (low energy high sensitivity) collimator has an opening diameter ofabout 2.54 mm, a length of about 24.04 mm, with an acceptable incidentangle range of 0° to about 6.03°. In some other examples, acceptableincident angle ranges from 0° to about 15°, or 0° to about to 75°. Forexample, an opening with a diameter of about 1.0 mm and a length ofabout 3.0 mm has an acceptable incident angle of about 18.43°, and anopening with a diameter of about 3.0 mm and a length of about 1.5 mm hasan acceptable incident angle of about 63.43°.

Between radiation source 152 and detector 102, collimator 104 functionsas a mask. Each radiation source 152 from patient 150 projects ontodetector 102 a shadow of the mask weighted by the intensity of radiationsource 152. The location of the shadow depends on the direction of theincident photons from radiation source 152. As a result, a raw imageacquired by detector 102 is a summation of the shadows cast by all ofthe radiation source 152 within field-of-view (FOV) of the collimator104. Image reconstruction is thus to construct an estimated image ofradiation sources 152 (referred to as object image) from the rawimage(s) acquired by detector 102 (referred to as measured image).Mathematically, an imaging system can be approximately represented by asimplified model without considering noise asp=f*h  (1)where * represents the convolution operator, p is the measured image, fis the object image, and h represents the coded pattern formed in thecollimator. In other words, h is a matrix, representing the coded maskshadow. In some embodiments, his a matrix of “0” s and “1” s, where eachelement corresponds to a grid position on the collimator, 1 representsan opening (e.g., a pinhole) at that position, and 0 representsotherwise. This matrix can be magnified to represent the magnifyingeffect of the mask shadow projected by a source on the detector, andinterpolation may be used to calculate the matrix element values.

Image reconstruction can be considered as a decoding procedure. Theobject image f can be estimated by decoding, i.e., correlating themeasured image p with a decoding mask pattern g. This decoding procedurecan be represented as{circumflex over (f)}=p⊗g=f*(h⊗g)  (2)where {circumflex over (f)} denotes an estimation of the object image, ⊗represents the correlation operator, and g is the decoding masksatisfyingh⊗g=δ  (3)

From a two-dimensional (2D) point of view, measured image p can bedivided into a plurality of smaller regions, representing by p_(i). Forexample, if based on x- and y-coordinates of the plane a collimatorresides, a region can be denoted as p_(xy). When a planar detector isused, the X- and Y-axes are usually parallel to the detector surface,and Z-axis is perpendicular to the detector surface. Similarly, objectimage f can be divided into a plurality of smaller regions, representingby f_(j) or as f_(xy) on an X-Y plane. In one embodiment, an imagereconstruction method is called a Maximum Likelihood Expectation andMaximization (MLEM) method. The MLEM method is derived from the Poissonnoise model and maximization of P({circumflex over (f)}|p), theprobability of an estimated object image {circumflex over (f)} given themeasured image p. The MLEM method estimates object images using

$\begin{matrix}{{\hat{f}}_{j}^{({k + 1})} = {\frac{{\hat{f}}_{j}^{(k)}}{\sum\limits_{i = 1}^{I}K_{ij}}{\sum\limits_{i = 1}^{I}\frac{p_{i}K_{ij}}{\sum\limits_{j = 1}^{J}{K_{ij}{\hat{f}}_{j}^{(k)}}}}}} & (4)\end{matrix}$where {circumflex over (f)}_(j) ^((k+1)) is the (k+1)th estimate of thejth element of f, the object image; p_(i) is the ith element of themeasured image, and K_(ij) is the transition matrix representing theprobability of photon emitted from j_(th) element of the object beingdetected by the ith element of the detector. The values of Kij can bedetermined by measurement, simulation, or modeling of the collimator.Let p_(r) be the denominator inside the second summation, p_(r)=Σ_(j=1)^(j)K_(ij){circumflex over (f)}_(j) ^((k)), representing an expectationof measured image based on the kth estimated object image, {circumflexover (f)}^((k)). This step is often referred to as the forwardprojection.

In some cases, such as when a coded aperture mask is used as acollimator, the projection image can be corrected for the angular effect(including cos³(θ) term and aperture collimation effect term) prior tothe MLEM iterative deconvolution process. By doing so, the imaging modelbecomes a convolution process and equation (4) can be further written as

$\begin{matrix}{{\hat{f}}^{({k + 1})} = {{\hat{f}}^{(k)} \times \left( {h \otimes \frac{p_{c}}{{\hat{f}}^{(k)}*h}} \right)}} & (5)\end{matrix}$whereas discussed above, h is a coded aperture mask shadow, and p_(c) isa measured image after correction for angular and collimation effects.Specifically, p_(c)=p/Cc, where Cc is the angular effect correctionfactor (e.g., in a matrix form) that is a smooth function that accountsfor one or more factors including cos³(θ) term and aperture collimationeffect term. Here the angle θ is between an incident path of a photonpassing through a point on collimator, usually the center of thecollimator, hitting a pixel on detector with respect to verticaldirection Z that is the normal of the top surface of collimator. And *and ⊗ represent convolution and correlation operations, respectively.Equation (5) includes two major procedures: forward projection andbackward projection. The convolution represents the forward projectionstep using current estimation of {circumflex over (f)}^((k)). Thecorrelation with the Cc correction resulting from the division step ofp_(c) represents the backward projection step.

Equation (5) is suitable for a thin imaging object whose thickness ismuch smaller than the distance between the subject and a collimator suchas collimator 122. For thicker objects, a three-dimensional (3D) methodis used. For example, an object image at distance z (measured from theplane where the detector resides to the object) can be estimated usingthe following equation:

$\begin{matrix}{{{\hat{f}}^{({K + 1})}(z)} = {\frac{{\hat{f}}^{(K)}(z)}{\sum\limits_{x,y}{h(z)}}\left\lbrack {{h(z)} \otimes \frac{p_{c}}{\sum\limits_{z}{{{\hat{f}}^{(K)}(z)}*{h(z)}}}} \right\rbrack}} & \left( {5 - 1} \right)\end{matrix}$

A slightly different formula can be used to estimate the subject aswell:

$\begin{matrix}{{{\hat{f}}^{({K + 1})}(z)} = {\frac{{\hat{f}}^{(K)}(z)}{\sum\limits_{x,y}{h(z)}}\left\lbrack {{h(z)} \otimes \frac{p_{c} - {\sum\limits_{z \neq z^{\prime}}{{{\hat{f}}^{(K)}\left( {z\prime} \right)}*{h\left( {z\prime} \right)}}}}{{{\hat{f}}^{(K)}(z)}*{h(z)}}} \right\rbrack}} & \left( {5 - 2} \right)\end{matrix}$where {circumflex over (f)}^((K))(z) is an estimation of the object atslice z after k iterations, and h(z) is the coded aperture mask shadowcorresponding to z. The process expressed in equation (5-2) differs fromthe conventional MLEM deconvolution in that the expected contributionfrom the “out-of-focus” slices (z′≠z) is subtracted from the measuredprojection, and the correction ratio in the division step is calculatedonly for the “in-focus” slice (z′=z). More specifically, the correctionratio is computed only from the estimation errors in the in-focus slice.Hence, the algorithm is expected to converge faster. Further regardingdistance z in 3D imaging, the partition of the object along Z-axis(perpendicular to the detector surface) may be equally spaced.Alternatively, the partition of object along Z-axis may be unevenlyspaced (e.g., of variable distances).

In one embodiment, an image reconstruction method may divide objectimage into smaller regions. For example, an object plane at height z maybe divided into n×n small regions, f_(i)(z), i=1, . . . , n². For theith region, a center angular effect correction factor C_(Ci) for thatregion was computed by taking the center of the region as thecollimation center. Then, in the forward projection step, thecontribution of the current estimated image plane to the projection,p_(r)(z), was calculated. More specifically, the {circumflex over(f)}^((K))(z)*h(z) originally formulated in equation (5-2) is computedas:p _(r)(z)=Σ_(i=1) ^(n) ² [{circumflex over (f)} _(i) ^((K))(z)*h(z)]×C_(Ci)  (6)where C_(Ci) is the angular effect correction factor for the ith region,C_(Ci)=C_(C) (x−x_(ci), y−y_(ci)), and x_(ci), y_(ci) are the x- andy-coordinates of a point in the ith region, usually the center of ithregion of {circumflex over (f)}^((K))(z). Note that all variables inequation (6) have x,y as parameters which are ignored for simplicity.Then the object image can be estimated using the following equation:

$\begin{matrix}{{{\hat{f}}^{({K + 1})}(z)} = {\frac{{\hat{f}}^{(K)}(z)}{\sum\limits_{x,y}{h(z)}}\left\lbrack {{h(z)} \otimes \frac{p - {\sum\limits_{z \neq z^{\prime}}{{\hat{f}}^{(K)}{p_{r}\left( {z\prime} \right)}}}}{p_{r}(z)}} \right\rbrack}} & \left( {6 - 1} \right)\end{matrix}$

A slightly different formula can be used to estimate the object as well:

$\begin{matrix}{{{\hat{f}}^{({K + 1})}(z)} = {\frac{{\hat{f}}^{(K)}(z)}{\sum_{x,y}{h(z)}}\left\lbrack {{h(z)} \otimes \frac{p}{\sum\limits_{z}{p_{r}(z)}}} \right\rbrack}} & \left( {6 - 2} \right)\end{matrix}$

Note that in both equations (6-1) and (6-2), the original measuredimage, p, is used instead of p_(c) as in equation (5).

The collimator matrix h described above assumes that recorded signal inp reflects only photons that can only pass through the openings on thecollimator. In reality, signals from other channel exist. For example,photons can penetrate the metal plate at a rather low but greater thanzero rate. For instance, with a 1 mm thick plate made of tungsten, lessthan 3% of photons with 140 keV energy can pass through. And there arerandom thermal or electrical events on the detector that contribute tothe signal such as dark current. Therefore, a thorough computation ofp_(r) would include computation of signals going through channels otherthan the holes. Nevertheless, signals from photons passing through holesrepresent significant portion of the total signal and are a majorcomponent of interest, and steps can be taken to minimize the signalsfrom other channels, such as increase plate thickness to reducepenetration.

The equations (5) and (6) described above are more accurate under asmall mask assumption, which means detector size is much larger than acoded aperture mask size. However, in many situations, a larger mask isused to increase detection sensitivity. Therefore, a small maskassumption does not always hold, resulting in degradation in the imagereconstruction.

To mitigate this issue and still approximate a small mask assumption, animage reconstruction method can be further improved by partitioning themask, optionally represented by collimator matrix h, into small regions,such that each region is small enough to satisfy the small maskapproximation. Thereafter, forward projection through each individualcollimator region is calculated. Center angular correction for thatcollimator region may also be applied to each forward projection foradjustment. The forward projection through each individual region isthen summed into an overall forward projection. For example, acollimator may be divided into n small regions, h_(i)(z), i=1, . . . ,n; for each region, a center angular effect correction factor C_(Ci)suitable for that region was computed, optionally by taking the centerof the region as the collimation center (Hereby the angle θ is betweenan incident path of a photon passing through a point in this collimatorregion, usually the center point of this region, hitting a pixel ondetector with respect to vertical direction Z that is the normal of thetop surface of collimator); then, in the forward projection step, thecontribution of the current estimated image plane to the projection wascalculated as p_(r)(z). More specifically, the {circumflex over(f)}^((K))(z)*h(z) originally formulated in equation (6) is computed as:p _(r)(z)=Σ_(i=1) ^(n) [{circumflex over (f)} ^((K))(z)*h _(i)(z)]×C_(Ci)  (7)where [{circumflex over (f)}^((K))(z)*h_(i)(z)] is the forwardprojection of the ith region of the collimator, C_(Ci) is the angulareffect correction factor for the ith region of the collimator, andp_(r)(z) is overall forward projection of image {circumflex over(f)}^((K))(z). In one example, if all openings of the collimator are ofthe same size and shape, then C_(Ci)=C_(C)(x−x_(ci), y−y_(ci)), which isthe same function C_(C) shifted by various (x_(ci), y_(ci)) values, andx_(ci), y_(ci) are the x- and y-coordinates of a point in the ith regionof h, h_(i), usually the center of that region. Even though each regionmay be expressed by the same function C_(C), each region may have itsown Cci value due to the shift in x- and y-coordinates, such that amongtwo different regions, the respective Cci values may be different. Ccivalue is also affected by through hole opening sizes and/or shapes. Inother words, if through hole opening sizes and/or shapes in two regionsare different, functions and values of Cci in these two regions may alsobe different. In some embodiments, the forward projections through atleast two regions overlaps. Note that formula (7) is an example and isnot limiting different ways to divide regions and calculate forwardprojection for each individual region. In some embodiment, the smallregions h_(i)(z) all contains a subset of openings of h, and at leastone of the small regions does not contain all openings of h. Forexample, in equation (7), {circumflex over (f)}^((K))(z) may be used asa whole, or being partitioned into small regions and summed together asin equation (6). In another example, frequency domain equivalence ofequation (7) may be used for calculation. In yet another example, one ormore individual regions may be omitted from equation (7), such as fortrading-off calculation speed as long as the accuracy level ofcalculation can still be satisfied.

FIGS. 3A and 3B illustrate a top view and corresponding matrix h of anexemplary collimator 104. The specific collimator illustrated in FIG. 3Ais a MURA 11 NTHT (no-two-holes-touching) pattern. A NTHT pattern is anextension of a basic pattern where a row of all zeros is insertedbetween every two adjacent rows and a column of all-zeros is insertedbetween every two adjacent columns of the basic pattern. As a result,the minimal hole-to-hole distance is at least two times of the holesize. The black dots in FIG. 3A represent the holes on the collimator104, which corresponds to the “one”s in the matrix h in FIG. 3B. For thematrix in FIG. 3B, there is an all-zeros column to the right withrespect to the holes in FIG. 3A which is part of the MURA 11 pattern.Dash lines present a way of partitioning collimator 104 into 4 regionsthat can be used in equation (7). In this specific example, collimator104 is divided into 2×2 regions. Also note that in this illustratedembodiment, each hole belongs to one and only one region after thepartition, and none of the regions has all of the holes. The partitionis not limited to 2×2 regions. Alternatively, collimator 104 can bedivided into regions of any suitable number, such as a×b regions, wherea and b represent any suitable integer, and all openings are dividedinto these regions and no openings appears in more than one region insome embodiments. Further, in some embodiments, a may be equal to b(a=b). In some alternative embodiments, a may not be equal to b (a≠b).In FIG. 3A, ‘+’ marks are overlay on collimator 104 to represent theposition (x_(ci), y_(ci)) used to calculate Cci for each region. In theillustrated embodiment, position (x_(ci), y_(ci)) is selected at thecenter of each region.

FIGS. 4A-4C illustrate various embodiments in partitioning regions on acollimator. Holes on collimator are omitted in FIGS. 4A-4C forsimplicity. Each region may independently have its own shape and size.In some embodiments, each region does not overlap with any neighboringregion. In some embodiments, two or more regions may have overlapping,such as to simplify a partition process, as long as the accuracy levelof calculation can still be satisfied. In FIG. 4A, each region has ahexagon shape and the same size. In FIG. 4B, the regions are acombination of squares and rectangles with different sizes. Further,regions may be abutted with or separated from neighboring regions. FIG.4C shows circular regions spaced from each other. In some embodiments,each region has a convex shape (no inner corner larger than 180degrees). In some embodiments, each region has at least one hole. If aregion has no holes or has only holes that no substantial radiation fromthe field of view would pass through, then it has little or nocontribution to the calculation and hence can be ignored. But it is notnecessary for each region to have same number of holes. Some regions mayhave more holes than others, or vice versa. Further, holes in the sameregion may have the same shape, but different between regions. Forexample, holes in square regions in FIG. 4B may be circular, while holesin rectangular regions in FIG. 4B may be square. In one embodiment, theholes in one or more of regions form coded aperture patterns, such asMURA patterns. The regions may be of different sizes, and the codedaperture patterns in the regions may be in different size, shape, ororientations. Furthermore, holes in the same region may have the samesize, but different between regions. In one embodiment, the hole sizesin the central regions may be larger (wider acceptance angle), andbecome smaller toward the peripheral regions. For regions havingdifferent hole sizes, corresponding angular effect correction factorsCci also have different function and thereby different values asexplained above. In other words, Cci may not be a constant but dependingon regions. In furtherance of some embodiments, even within the sameregion, there may be holes of different sizes and/or shapes. Acalculation of Cci of such a region may include first calculatingdifferent functions corresponding to different hole sizes and/or shapes,and then calculating an average of these different functions as a valuefor Cci. As an example, the calculation of the average may includeweighing each function based on its corresponding holes' amountpercentage in the total number of holes then summing the weightedfunctions.

It is worth to note that the computation of p_(r) in equation (7) mayinclude other elements such as penetration and background noise.However, the contribution of the above formula is a major component ofp_(r) which account for more than half of the signal strength. Sinceconvolution and correlation are mirror operations, the followingequation computing backward projection (or back projection) fromrespective forward projection can be used in the computationsq(z)=Σ_(i=1) ^(n) [r(z)⊗h _(i)(z)]×C _(Ci)  (7-1)where r(z)⊗h_(i)(z) is the backward projection of the ith region of thecollimator, C_(Ci) is the angular correction for the ith region. Bycomputing backward projection of each region, q(z) as overall backwardprojection can be computed as a sum of the backward projections of eachregion. And r(z) can be the quotient inside the bracket in equations(7-1) or (7-2).

$\begin{matrix}{{r(z)} = {\frac{p - {\sum\limits_{z \neq z^{\prime}}{p_{r}\left( {z}^{\prime} \right)}}}{p_{r}(z)}{or}}} & \left( {7 - 2} \right)\end{matrix}$ $\begin{matrix}{{r(z)} = {\frac{p}{\sum\limits_{z}{p_{r}(z)}}.}} & \left( {7 - 3} \right)\end{matrix}$Either equation (7-2) or (7-3) can be considered as comparing themeasured image p with forward projections. The partition of h used inequation (7-1) for computing backward projection may or may not be thesame as the partition of h used in equation (7) for computing forwardprojection. For example, to compute backward projection, h may bepartitioned into more regions than when it is partitioned to computerespective forward projection. Alternatively, to compute backwardprojection, h may be portioned into regions of different shapes thanwhen it is portioned to compute respective forward projection. In anexample, frequency domain equivalence of equation (7-1) may be used forcalculation. In another example, one or more individual regions hi maybe omitted from equation (7-1), such as for trading-off calculationspeed as long as the accuracy level of calculation can still besatisfied.

Note that the method is an iterative method where {circumflex over(f)}^((k+1)) is the (k+1)th estimate of the object image. The iterationin eq. (5) can be simplified as{circumflex over (f)} ^((k+1))(z){circumflex over (f)}^((k))(z)×q(z)  (7-4)The value of q(z) may first be normalized before calculating {circumflexover (f)}^((k+1)).

It is worth to note that forward projections can be used in otherapplications such as simulation, i.e., generating simulated projectionsfor a given or virtual object, their calculations can be done through(7) as well. The partition of h does not have to be same for alliterations. The number of regions to divide h into is a considerationtrading off reconstructed image quality and computation speed. Theregions in a partition can be very small, and may even contain only onehole. Smaller regions with fewer holes represent better approximation tothe small angle approximation and higher precision, but may slow downcomputation speed, while larger regions with more holes may increasecomputation speed but with poorer image quality. The smaller the regionh_(i) is, it is usually more accurate, but the computational complexityis higher as well. Therefore, the partition can start coarsely (smallernumber of regions) in the beginning for faster computation, and finerpartitions (larger number of regions) can be used as iterationsprogress.

In SPECT imaging, multiple planar images are acquired from differentangles by rotating camera around the object (e.g., patient 150). Whencoded aperture mask is used for collimation, a 2D or 3D image can bereconstructed from one planar image, such as applying equations (6) and(7). Note that in equations (6) and (7), p or p_(c) are used, whichrepresents one planar image, meaning f is estimated using one acquiredplanar image. The iterations in eq. (4), (5)-(5-2), can all besimplified as{circumflex over (f)} _(j) ^((k+1))(z)={circumflex over (f)} _(j)^((k))(z)×q _(j)(z)  (8)where q_(j) is the updating factor for jth element of f(z).

This invention also presents methods to reconstruct object image usingmore than one acquired planar images, which may be acquired fromdifferent angle, or with different distance from the object, ordifferent distance of collimator to detector, or with shift, tilting ofcamera(detector) or object, or a combination of two or more of theabove. Because more data are used to reconstruct the image, the accuracyand signal-to-noise ratio may be higher. When more than one acquiredplanar images are used, the updating factor can be

$\begin{matrix}{{q_{j}(z)} = {\sqrt[m]{\sum\limits_{t = 1}^{T}{{q_{j}^{m}\left( {z,t} \right)}/T}}{or}}} & (9)\end{matrix}$ $\begin{matrix}{{q_{j}(z)} = \sqrt[T]{\prod\limits_{t = 1}^{T}{q_{j}\left( {z,t} \right)}}} & (10)\end{matrix}$where T is the number of projections used, and m is a positive number,and q_(j)(z, t) is the updating factor for element j of f(z) calculatedby the method presented in this invention using the tth acquired imagealone. Note that when the underlying grid of the coordinate systemq_(j)(z, t) is different from each other (i.e., for different t),interpolation and/or rotation/flip may be performed so that all of themare defined on the same grid.

A special case of an embodiment of this method is to update f using twoopposite projections. In such scenario, since the slices in the 3Dimages are in X-Y plane and parallel to the detector surface in bothacquisition positions, only axis flip is needed but no rotationcomputation is required in calculating equation (9) or (10). Since manyclinical SPECT systems employ two cameras mounted at opposite positionswhere two opposite projections are acquired simultaneously, thisembodiment has the advantage of processing the pairs of two oppositeprojections as they are acquired, streamlining the data flow. Anotherspecial case of the embodiment of this method is to update fusing twoopposite projections (images acquired with 180 degrees apart), and twoprojections perpendicular to the previous two opposite projections.Since the slices in the 3D images are parallel to the detector surface,the axes of these orthogonal projections are aligned with some axispermutation, no rotation may be needed in calculating equation (9) or(10).

Referring now to FIG. 5 , a flow chart of a method 500 for imagereconstruction for collimator and detector based medical imaging systemsis illustrated according to various aspects of the present disclosure.The method 500 is merely an example and is not intended to limit thepresent disclosure to what is explicitly illustrated in the method 500.Additional operations can be provided before, during, and after themethod 500, and some operations described can be replaced, eliminated,or moved around for additional embodiments of the method. The method 500is described below in conjunction with FIGS. 1-4 .

At operation 510, an object (e.g., patient 150) and an imaging apparatus(e.g., imaging apparatus 100) is provided. The imaging apparatusincludes a collimator (e.g., collimator 104) and a detector (e.g.,detector 102). The collimator is disposed between the object and thedetector. At operation 520, the collimator filter photons emitted fromthe object and the detector acquire a measured image from the photonsarriving at the detector. The imaging apparatus may rotate around theobject to acquire multiple measured images from multiple angles. In oneexample, the imaging apparatus acquires measured images from 0°, 90°,180°, and 270° angles surrounding patient 150.

At operation 530, an initial estimation of the object image ({circumflexover (f)}⁽⁰⁾) is provided as the starting point of the iterationprocess. A proper initial guess may be used for {circumflex over(f)}⁽⁰⁾, such as a matrix of all “one”s. In another example, a ComputedTomography (CT) image may be used for {circumflex over (f)}⁽⁰⁾. Modernmolecular imaging systems are often delivered in hybrid form with CTscanners in tandem, such as a hybrid SPECT/CT or PET/CT systems. CTimages are created with X-ray passing through an object, such as apatient lying on a couch undergoing SPECT or PET imaging. When a CT scanis available prior to SPECT reconstruction, the CT scan can be used togenerate the initial guess. In one embodiment, the CT scan can be usedas a finite support, i.e, the contour of the patient body can be used todefine the scope of “one”s in the matrix mentioned above, setting thematrix element values to be “one” inside the patient contour and “zero”if outside.

The method 500 then moves to operation 540 to start an iterationprocess. The iteration process (also includes operations 550-580) may beexecuted by the computer system 130, such as by the image processor 134.At operation 540, the collimator is partitioned into a plurality ofregions. Each region has at least one hole. A matrix corresponding tothe collimator is also partitioned accordingly.

At operation 550, the method 500 calculates a forward projection from anestimation of the object image and a partitioned matrix representingeach region, such as from a convolution using equation (7) or an MLEMmethod using equation (7-4) in various embodiments. The forwardprojection is also adjusted by an angular effect correction factor(s) ofeach region including a cos³(θ) term and aperture collimation effectterm.

At operation 560, the method 500 calculate a backward projection fromthe forward projection result from previous operation, such as by usinga correlation operation described in equation (7-1). The backwardprojection is also adjusted by an angular effect correction factor ofeach region including a cos³(θ) term. Calculating backward projectionmay use the same partition as the one used in respective forwardprojection, or alternatively may use a different partition, such as withdifferent numbers and/or shapes of regions. Operation 560 may furtherinclude intermediate steps, such as calculating r(z) using p andp_(r)(z) in equations (7-2) and (7-3). Further, r(z) may be interpolatedto different grid size. Subsequently r(z) is used to calculate backwardprojection.

At operation 570, the method 500 reconstructs the object image using thecalculated backward projection from previous operation, such as usingequation (8). Method 500 then determines whether iteration may end, suchas by predetermined iteration steps or when difference off betweenconsecutive iterations are below a predetermined threshold (e.g.,<0.5%). If the iteration does not end, then the {circumflex over(f)}^((k)) will be used for next iteration as an input for operation 540to calculate {circumflex over (f)}^((k+1)). Operation 540 mayrepartition the mask to get finer meshing (e.g., more number of regions)for higher accuracy. If the iteration ends, method 500 proceeds tooperation 580 that the estimated object image is stored (e.g., in imagestorage 136) and/or displayed (e.g., on display 138).

Dividing a collimator into multiple regions and calculating forward andbackward projections associated with respective individual regionsprovide a possibility of a novel collimator design. In one example, whenthe collimator is represented in matrix form, h, less than 70% of theelements are ones. More specifically, for example, for common codedaperture collimators such as those in URA and MURA arrays, the number ofones is less than 50% of total number of elements. An exemplarycollimator design is illustrated in FIG. 6 . As discussed above,collimator can be considered as a metal plate with a plurality of holes,such as through holes. In FIG. 6 , a black dot represents a hole. Intraditional coded aperture design, the holes on the whole collimatorform one specific coded pattern, such as a URA array or a MURA array. Asa comparison, the collimator in FIG. 6 is divided into a plurality ofregions (or groups) (e.g., regions I, II, III, and IV), and throughholes in each region by themselves independently form a coded aperturepattern, including but not limited to: URA array, MURA array, randomarray, and pseudo random array. In a specific embodiment, each regionhas a unique pattern selected from either a URA array or a MURA array.The pattern may not be the same for all groups, the hole sizes andshapes may also be different, or the orientation of rows or columns ofholes in the individual groups may be different. An exemplary collimatorwith a region of different orientation is illustrated in FIG. 7 . InFIG. 7 , compared with patterns in regions II, III, and IV, the patternin region I has an orientation with respect to a normal direction to thetop surface of the collimator. This orientation is denoted as an angleβ. In various embodiments, the angle β may be in a range from about 5degrees to about 85 degrees, such as 45 degrees. The holes can beregarded as landing on a meshing grid on the collimator. In someembodiments, a grid in one region may be different than those in otherregions. For example, one region may have larger grid size than others,or smaller vice versa. The illustrated collimator in FIG. 7 can also beregarded as having a different grid in region I than others. Althoughthe collimators illustrated in FIGS. 6 and 7 have four groups of codedaperture patterns, in various embodiments, a collimator may have othernumber of groups, such as two, three or more than four groups. In aparticular example, a collimator may have 9 groups of coded aperturepatterns in a 3×3 array.

The holes in different groups on the same plate can be of differentsizes and different shapes. Furthermore, the holes in the same group canbe of different sizes and different shapes. Since the incident angle θ(FIG. 2B) is generally defined by the hole size and length, differentholes in the same collimator may have different incident angles. In someembodiments, the minimum incident angle θ of all the holes in thecollimator is at least 10°, such as at least 15° in some specificexamples, which has the benefit of a larger FOV. The FOV of each groupmay overlap with at least one other group on the same plate (within thedesigned range of imaging: usually the FOV is bigger when it is fartheraway from the collimator or aperture). In some embodiment, the distancebetween the groups (the minimum hole-to-hole distance between twodifferent groups) is larger than the minimal hole-to-hole distancewithin each group, may be more than twice the minimal hole-to-holedistance within each group, for example at least 2.5 times the minimalhole-to-hole distance within each group. In one embodiment, the minimalhole-to-hole distance within each group is less than 2 times the holesize. Hole size is defined as a size of smallest square that can fullycovers the hole. Hole-to-hole distance is a pitch defined by thedistance from the center of one hole to the center of an adjacent hole.In some embodiments, the number of holes can be larger than 30. Forexample, in each region, there can be one of 12, 24, 60, 84, 144, 180,and 264 holes in a MURA 5, 7, 11, 13, 17, 19 and 23 patterns. In someother embodiments, a collimator mask may have a mosaic of 4 identicalMURA coded aperture patterns organized in a 2-by-2 form with theseparation between the neighboring holes from different groups being thesame as the spacing between holes of the same group. The four patternsare identical in size, shape, and orientation. The reconstruction methodpresented in this invention is also applicable to this type ofcollimators by partitioning the holes in the groups, including a 2-by-2partition. Further, in some embodiments, the density of the holes in thecollimator is less than 50%, typically less than 35% or 30%. And in someembodiments, the density of the holes in the collimator is greater than2%, typically greater than 5% or 7.5%. The density of the holes in thecollimator is defined as the total area of the holes (total opening)over the area of the smallest convex polygon that can surround all theholes. In case of hole shapes other than straight through holes, such asknife-edge or channel-edge holes, the smallest section of the hole areais used in the calculation of total hole area. Similarly, in someembodiments, the density of the holes in each region is less than 50%,typically less than 35% or 30%. And in some embodiments, the density ofthe holes in each region is greater than 2%, typically greater than 5%or 7.5%. The density of the holes in a region is defined as the totalarea of the holes (total opening) in that region over the area of thesmallest convex polygon that can surround all the holes in that region.Similarly, in case of hole shapes other than straight through holes,such as knife-edge or channel-edge holes, the smallest section of thehole area is used in the calculation of total hole area.

Although not intended to be limiting, one or more embodiments of thepresent disclosure provide many benefits for molecular imaging of asubject such as a patient. For example, the image reconstruction methodsincrease imaging sensitivity and resolution even when a relatively largecollimator not meeting small mask assumption is used. Therefore, systemperformance is improved.

The foregoing outlines features of several embodiments so that those ofordinary skill in the art may better understand the aspects of thepresent disclosure. Those of ordinary skill in the art should appreciatethat they may readily use the present disclosure as a basis fordesigning or modifying other processes and structures for carrying outthe same purposes and/or achieving the same advantages of theembodiments introduced herein. Those of ordinary skill in the art shouldalso realize that such equivalent constructions do not depart from thespirit and scope of the present disclosure, and that they may makevarious changes, substitutions, and alterations herein without departingfrom the spirit and scope of the present disclosure. Accordingly, it isappropriate that the appended claims be construed broadly and in amanner consistent with the present disclosure.

What is claimed is:
 1. A method of imaging reconstruction, comprising: providing a target object; acquiring measured images of the target object, wherein each of the measured images is acquired by filtering radiation from the target object by a mask having multiple holes and detecting filtered radiation by a detector; providing an estimated image of the target object; for each of the measured images, calculating an updating factor, thereby obtaining multiple updating factors, wherein the calculating of the updating factor includes: partitioning a mathematical representation of the mask used for acquiring the respective measured image into multiple first regions such that each hole of the mask belongs to one and only one first region and none of the first regions has all of the holes; for each of the first regions of the mathematical representation of the mask, deriving a separate forward projection from the estimated image of the target object and the respective first region, thereby acquiring multiple forward projections, wherein the multiple forward projections have one-to-one correspondence with the multiple first regions; and comparing the respective measured image of the target object with the forward projections; and updating the estimated image of the target object based on the updating factors.
 2. The method of claim 1, further comprising performing an operation to the updating factors so that the updating factors are aligned to a common underlying grid.
 3. The method of claim 2, wherein the operation is one of an interpolation, flip, or rotation operation.
 4. The method of claim 1, wherein at least two of the measured images are acquired at different angles relative to the target object.
 5. The method of claim 4, wherein the at least two of the measured images are acquired at opposite directions relative to the target object.
 6. The method of claim 5, wherein another two of the measured images are acquired at a direction that is perpendicular to the opposite directions the at least two of the measured images are taken from.
 7. The method of claim 1, wherein at least two of the measured images are acquired at different distances from the target object.
 8. The method of claim 1, wherein at least two of the measured images are acquired with different distances between the detector and the mask that are used to acquire the at least two of the measured images.
 9. The method of claim 1, wherein the calculating of the updating factor further includes: deriving a backward projection based on a result of the comparing the respective measured image of the target object with the forward projections, wherein the deriving of the backward projection includes partitioning the mathematical representation of the mask used for acquiring the respective measured image into multiple second regions such that each hole of the mask belongs to one and only one second region and none of the second regions has all of the holes.
 10. The method of claim 9, wherein the multiple second regions and the multiple first regions have different number of regions or different shapes of regions.
 11. A method of imaging reconstruction, comprising: providing a target object, a detector, and a mask disposed between the target object and the detector, wherein the mask includes multiple holes; acquiring a measured image of the target object by the detector; providing an estimated image of the target object; first partitioning a mathematical representation of the mask into multiple first regions such that each hole belongs to one and only one first region and none of the first regions has all of the holes; for each of the first regions of the mathematical representation of the mask, deriving a forward projection from the estimated image of the target object and the respective first region, thereby acquiring multiple forward projections, the multiple forward projections having a one-to-one correspondence with the multiple first regions; comparing the measured image of the target object with the forward projections; second partitioning the mathematical representation of the mask into multiple second regions such that each hole belongs to one and only one second region and none of the second regions has all of the holes; for each of the second regions of the mathematical representation of the mask, deriving a backward projection based on a result of comparing the measured image of the target object with the forward projections, thereby obtaining multiple backward projections, the multiple backward projections having one-to-one correspondence with the multiple second regions; and updating the estimated image of the target object based on the multiple backward projections.
 12. The method of claim 11, wherein the multiple first regions are the same as the multiple second regions.
 13. The method of claim 11, wherein there are more regions in the multiple second regions than in the multiple first regions.
 14. The method of claim 11, wherein the multiple second regions have a different shape than the multiple first regions.
 15. The method of claim 11, wherein the deriving of the forward projection includes for each of the first regions calculating a respective first angular effect correction factor, and the deriving of the backward projection includes for each of the second regions calculating a respective second angular effect correction factor.
 16. A medical imaging system, comprising: one or more collimators configured to filter radiation emitted from a target object; one or more detectors each configured to acquire a measured image of the target object by detecting the radiation that is filtered by one of the one or more collimators; and a controller operable to execute computer-readable codes to perform the following operations: receiving first and second measured images from the one or more detectors, wherein the first and second measured images are taken from opposite directions; providing an estimated image of the target object; first partitioning a mathematical representation of the collimator into multiple first regions; for each of the first regions of the mathematical representation of the collimator, deriving a forward projection for each of the first and second measured images, thereby obtaining multiple forward projections for each of the first and second measured images, wherein the multiple forward projections have one-to-one correspondence with the multiple first regions; first comparing the first measured image with the multiple forward projections derived for the first measured image; second comparing the second measured image with the multiple forward projections derived for the second measured image; and updating the estimated image based on the first comparing and the second comparing.
 17. The medical imaging system of claim 16, wherein the controller is further operable to execute computer-readable codes to perform the following operations: receiving third and fourth measured images from the one or more detectors, wherein the third and fourth measured images are taken from directions perpendicular to the opposite directions the first and second measured images are taken from; for each of the first regions of the mathematical representation of the collimator, deriving a forward projection for each of the third and fourth measured images, thereby obtaining multiple forward projections for each of the third and fourth measured images, wherein the multiple forward projections have one-to-one correspondence with the multiple first regions; third comparing the third measured image with the multiple forward projections derived for the third measured image; and fourth comparing the fourth measured image with the multiple forward projections derived for the fourth measured images, wherein updating the estimated image is further based on the third comparing and the fourth comparing.
 18. The medical imaging system of claim 16, wherein the controller is further operable to execute computer-readable codes to perform the following operations: second partitioning the mathematical representation of the one or more collimators into multiple second regions; and for each of the second regions of the mathematical representation of the one or more collimators, deriving a backward projection for each of the first and second measured images based on the first comparing and the second comparing, respectively, thereby obtaining multiple backward projections for each of the first and second measured images, wherein the updating is based on the multiple backward projections for each of the first and second measured images.
 19. The medical imaging system of claim 16, wherein the first comparing and the second comparing produce two updating factors, and wherein the controller is further operable to execute computer-readable codes to perform: performing a flip operation to one of the two updating factors.
 20. The medical imaging system of claim 16, wherein the first and the second measured images are acquired by two different detectors. 